Lappeenranta University of Technology Department of Mechanical Engineering. Dr.Tech.,Professor Aki Mikkola Dr.Tech.,Docent Asko Rouvinen

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1 Lappeenranta Unversty o Technology Department o Mechancal Engneerng The Dynamc Analyss o a Human Skeleton Usng the Flexble Multbody Smulaton Approach The topc o the Thess has been conrmed by the Department Councl o the Department o Mechancal Engneerng on 1 st November, 26. Supervsors: Dr.Tech.,Proessor Ak Mkkola Dr.Tech.,Docent Asko Rouvnen Lappeenranta, 9 th November, 26 Ram Abdullah Mohammad Al Nazer Vänölänkatu 27 B 14 Lappeenranta 531 Fnland Tel: ram.al.nazer@lut.

2 ABSTRACT Author: Ram Al Nazer Ttle: The Dynamc Analyss o a Human Skeleton Usng the Flexble Multbody Smulaton Approach Department: Mechancal Engneerng Place: Lappeenranta Year: 26 Master s thess. Lappeenranta Unversty o Technology 58 pages, 33 gures and 6 tables Supervsors: Dr.Tech.,Proessor Ak Mkkola and Dr.Tech.,Docent Asko Rouvnen Keywords: Flexble multbody dynamc, human lower lmb model, bone strength The human moton study, whch reles on mathematcal and computatonal models n general, and multbody dynamc bomechancal models n partcular, has become a subject o many recent researches. The human body model can be appled to derent physcal exercses and many mportant results such as muscle orces, whch are dcult to be measured through practcal experments, can be obtaned easly. In ths work, a human skeletal lower lmb model consstng o three bodes s bult usng the lexble multbody dynamc smulaton approach. The loatng rame o reerence ormulaton s used to account or the lexblty n the bones o the human lower lmb model. The man reason o consderng the lexblty n the human bones s to measure the strans n the bone result rom derent physcal exercses. It has been perceved that the bone under stran wll become stronger n order to cope wth the exercse. On the other hand, the bone strength s consdered an mportant actor n reducng the bone ractures. The smulaton approach and model developed n ths work are used to measure the bone stran results rom applyng rasng the sole o the oot exercse. The smulaton results are compared to the results avalable n lterature. The

3 comparson shows good agreement. Ths study sheds the lght on the mportance o usng the lexble multbody dynamc smulaton approach to buld human bomechancal models, whch can be used n developng some exercses to acheve the optmal bone strength.

4 ACKNOWLEDGEMENTS Ths research was carred out n the Department o Mechancal Engneerng, Lappeenranta Unversty o Technology, between May 26 and November 26. Ths thess would not have been completed wthout the support o my great supervsor, Ak Mkkola, a Proessor n the Mechatroncs and Vrtual Engneerng Laboratory. Although I have worked hard to complete ths thess, the nal product came about as a result o hs enthusastc overtures. Wth hm I had many wonderul and persuasve dscussons. To hm I owe more than words can descrbe. I am also grateul to the second supervsor o ths thess Dr. Asko Rouvnen or hs valuable comments and advce. I am especally thankul to my rend Mr. Tmo Rantalanen, a researcher n the department o bology o physcal actvty n Unversty o Jyvaskyla, or hs valuable comments and advce as well as enthusastc support at all stages o the work. I would lke also to thank my colleague at the Department o Mechancal Engneerng Mr. Tuomas Rantalanen or hs assstance and help. Last but no least, my amly has been always the source o love and support n my le. My ather Abdullah, s always the source o wsdom n my le. My mother Tagreed, orever I am grateul. My brothers Romel, Ramez and Mohammed and my sster Raya, wth your care and love you make the hard easy nront o me.

5 NOMENCLATURE Abbrevatons ADAMS Automatc Dynamc Analyss o Mechancal Systems CMS Component Mode Synthess CT Computerzed Tomography DAE Derental Algebrac Equatons DOF Degree o Freedom MRI Magnetc Resonance Imagng VIVO Latn expresson reers to expermentaton done on a lvng body Symbols A Aθ Transormaton matrx o the lexble body coordnate system Matrx results rom takng the dervatve o the transormaton matrx o the lexble body coordnate system wth respect to the rotaton angle ak j B Br C j C Cq D E j e Egenvectors correspond to the egenvalues o the lexble body Boolean transormaton matrx denes the vector o element j nodal coordnates n terms o the total vector o elastc nodal coordnates o the lexble body Boolean reerence transormaton matrx denes the reerence condtons appled to the lexble body Vector o lnearly ndependent constrant equatons o the multbody system 2 x 2 transormaton matrx denes the orentaton o the element j coordnate system wth respect to the lexble body coordnate system Jacoban matrx Stran-dsplacement matrx o the lexble body Symmetrc matrx o the elastc coecents o the lexble body Vector o the element j nodal coordnates

6 e j eo Vector o the elastc nodal coordnates o the lexble body Vector o the nodal coordnates o the element j n the undeormed state j e Vector o the nodal coordnates o the element j n the deormed state & e& F I I1 ~ I K K Acceleraton vector o the elastc nodal coordnates o the lexble body External orce actng on a lexble body 2 x 2 dentty matrx Vector denes the moment o the body mass about the axes o the lexble body coordnate n the undeormed state Skew symmetrc matrx Generalzed stness matrx assocated wth the generalzed coordnates o the lexble body Symmetrc postve dente stness matrx assocated wth the generalzed elastc coordnates o the lexble body K p Kˆ k M M p Mˆ m m md mˆ m Modal stness matrx o the lexble body Crag-Bampton stness transormaton matrx o the lexble body Modal stness coecents o the lexble body Mass matrx o the lexble body Modal mass matrx o the lexble body Crag-Bampton mass transormaton matrx o the lexble body Number o low requency mode shapes Mass o the lexble body Number o uncoupled derental equatons Modal mass coecents o the lexble body Constant submatrx o the mass matrx assocated wth the elastc lexble body coordnates

7 m RR m R Submatrx o the mass matrx assocated wth the translatonal coordnates o the lexble body coordnate system Submatrx o the mass matrx represents the couplng between the lexble body translatonal reerence moton and the elastc deormaton m Rθ Submatrx o the mass matrx represents the couplng between the lexble body reerence translatonal and rotatonal coordnates mθ Submatrx o the mass matrx represents the couplng between the lexble body deormaton rotatonal reerence moton and the elastc mθθ Constant submatrx o the mass matrx assocated wth the lexble body rotatonal reerence coordnates ( m θθ ) Scalar represents the change n the mass moment o nerta o the r lexble body due to deormaton ( m θθ ) Scalar represents the change n the mass moment o nerta o the rr lexble body due to deormaton ( m θθ ) Scalar represents the mass moment o nerta o the lexble body n Nˆ n nb the undeormed state about perpendcular axs to the plane Transormaton matrx transorms the modal coordnates o the Crag-Bampton modes o the lexble body to an equvalent orthogonal modal coordnates Number o generalzed coordnates o the multbody system Total number o the bodes n the planar multbody dynamc system nc n nn O j O P Number o the constrant equatons o the multbody system Number o the elastc coordnates o the lexble body Number o the elastc nodal coordnates o the lexble body Orgn o the lexble body coordnate system Orgn o the element j coordnate system Arbtrary pont on the lexble body

8 j P p p pr pˆ Arbtrary pont on element j Vector o the modal coordnates o the lexble body Vector o the modal elastc nodal coordnates o the lexble body Vector o the modal reerence coordnates o the lexble body Vector o the modal coordnates o the Crag-Bampton modes o the lexble body pˆ C Vector o modal coordnates o the Crag-Bampton constrant modes o the lexble body pˆ N Vector o modal coordnates o the Crag-Bampton xed boundary normal modes o the lexble body Q Qe Vector o the generalzed elastc and external orces o the lexble body Vector o generalzed external orces assocated wth the generalzed coordnates o the lexble body Q Vector o generalzed external orces assocated wth elastc coordnates o the lexble body QR Q v Qθ Vector o generalzed external orces assocated wth the translatonal reerence coordnates o the lexble body Quadratc velocty vector o the lexble body Vector o generalzed external orces assocated wth the rotatonal v ) reerence coordnates o the lexble body (Q Quadratc velocty vector assocated wth the elastc coordnates o v ) R the lexble body (Q Quadratc velocty vector assocated wth the translatonal reerence coordnates o the lexble body ( Q q v ) θ Quadratc velocty vector assocated wth the rotatonal reerence coordnates o the lexble body Vector o the total multbody system generalzed coordnates

9 q q q r q& q& q& & Generalzed coordnates system o the lexble body Vector o elastc coordnates o the lexble body Vector o reerence coordnates o the lexble body Velocty o the generalzed coordnates system o the lexble body Velocty o the generalzed elastc coordnates o the lexble body Acceleraton o the generalzed elastc coordnates o the lexble body R R & rp Vector o the translaton o the orgn o the lexble body coordnate system wth respect to the global coordnate Vector o the translatonal veloctes o the orgn o the lexble body wth respect to the global coordnate system Poston vector o any arbtrary pont on the lexble body wth respect to the global coordnate system j rp Poston vector or any arbtrary pont on the element j wth respect to the global coordnate system r& P & r& P Velocty vector o any arbtrary pont on the lexble body wth respect to the global coordnate system Acceleraton vector o any arbtrary pont on the lexble body wth respect to the global coordnate system S j S S ~ T t u u Space dependent shape matrx o the lexble body Space dependant shape matrx o the element j Constant skew symmetrc matrx o the lexble body Knetc energy o the lexble body tme Poston vector o any arbtrary pont on the lexble body wth respect to the body coordnate system Deormed poston vector o any arbtrary pont on the lexble body wth respect to the body coordnate system

10 uo Undeormed poston vector o any arbtrary pont on the lexble body wth respect to the body coordnate system j u V Vector o the assumed dsplacement eld o the nodal coordnates o the element j wth respect to the lexble body coordnate system Volume o the lexble body δ W e Vrtual work o external orces actng on a lexble body δ W s Vrtual work done by the elastc orces actng on a lexble body j w X 1 X 2 X X 1 X 2 j j 1 X 2 Vector o the assumed dsplacement eld o the nodal coordnates o the element j wth respect to the element j coordnate system Global coordnate system Flexble body coordnate system Element j coordnate system Greek Letters θ θ & ρ ωk Orentaton angle o the lexble body coordnate system wth respect to the global coordnate system Angular velocty o the lexble body coordnate system wth respect to the global coordnate system Densty o the lexble body Stress vector o the lexble body Stran vector o the lexble body Vector o Lagrange multplers Egenvalues or natural requences assocated wth each nodal coordnate o the lexble body Modal transormaton matrx Modal transormaton matrx or the total vector o the generalzed coordnates system o the lexble body Crag-Bampton transormaton matrx o the vector o generalzed coordnates system o the lexble body

11 IC Crag-Bampton transormaton matrx o the vector o elastc nodal coordnates o the lexble body or the neteror DOF n the constrant modes IN Crag-Bampton transormaton matrx o the vector o elastc nodal coordnates o the lexble body or the neteror DOF n the normal modes λ Egenvalues or natural requences assocated wth each modal coordnate o the Crag-Bampton modes o the lexble body Orthogonal egenvectors correspond to the egenvalues assocated wth each modal coordnate o the Crag-Bampton modes o the lexble body

12 Contents 1 INTRODUCTION Scope o the Work. 4 2 FLOATING FRAME OF REFERENCE FORMULATION Descrpton o Knematcs Knematc Constrants Descrpton o Inerta Generalzed Forces Quadratc Velocty Vector Reerence Condtons Equatons o Moton Fnte Element Assemblng Procedure Modal Reducton.22 3 SIMULATION MODEL OF THE HUMAN LOWER LIMB Shank Model Thgh Model Foot Model Assembly o the Model 36 4 NUMERICAL EXAMPLE Results and Dscusson CONCLUSION REFERENCES.57

13 1. INTRODUCTION Multbody dynamc approach s a mathematcal tool that can be used to model derent mechancal and structural systems. For nstance, such systems ncluded under the denton o multbody systems comprse robots, manpulators, vehcles and human skeleton. Fgure 1.1 llustrates a general multbody system shown n abstract orm. Jont External orces Body 1 Body 2 Sprng Jont Body 3 Body 5 Damper Body 4 Fgure 1.1. General multbody system. As depcted rom Fgure 1.1, a multbody system conssts o several bodes, whch can be rgd, lexble or combnaton o them. Those bodes are connected together by means o knematc jonts descrbed by constrant equatons. The orces appled over the multbody system bodes may be a result o sprngs, dampers, actuators or any external appled orces such as gravtatonal orces. The bomechancal human models are typcally more complcated than techncal multbody systems, as they nvolve a larger varety o jont types and body orms, complex actuators n the orm o muscles, connected groups o bones and

14 neghbourng sot tssue [1]. Recently, multbody dynamc approach has been used extensvely n modellng human skeleton. Several mportant results can be depcted rom the human skeleton model. For nstance, the nternal orces n the skeleton and muscular reactons. The key ssue rom usng multbody dynamc n human skeleton modellng s that there are no expermental methodologes capable o measurng these depcted parameters [2]. In other words, usng the multbody dynamc model o the human skeleton, several mportant parameters can be measured, whch are o a major mportance n many scentc elds such as, medcne, sports and bomechancal engneerng. In ths partcular study, a part rom human skeleton whch s the lower lmb shown n Fgure 1.2, s modelled usng multbody dynamc approach. Thgh Shank Foot Fgure 1.2. Lower lmb [3]. It can be notced rom Fgure 1.2 that the lower lmb conssts o several bodes connected to each other by means o jonts, wth external orces appled on these

15 bodes. Thereore by comparng the lower lmb shown n Fgure 1.2 wth the general multbody system shown n Fgure 1.1, one may notce that the multbody dynamc approach can be appled to model the lower lmb. For smplcty, the lower lmb model wll be consstng o three bodes; the thgh, the shank and the oot. The motvaton o ths study s to calculate the strans n the lower lmb bones. It has been perceved that the bone under stran wll become stronger n order to cope wth the exercses nduced loadng. The strength o the bone s consdered an mportant actor n reducng the rsk o bone osteoporotc racture, whch s manly, aectng the aged people. Thereore, many sport exercses can be developed, n order to mantan or ncrease the bone strength [4]. In order to calculate the strans n the lower lmb bone, the bone has to be modelled as a lexble body. Many prevous studes have been conducted n human skeleton modellng. In the work o Be and Fregly (24) a detaled musculoskeletal multbody model s created n order to predct muscle orces and contact pressures smultaneously n a knee jont [5]. In ths model, the lexble contact o the jont s combned wth the rgd body dynamcs o two bones. Slva et al. [6] have ncluded the natural boundary condtons or jonts n a multbody model n order to prevent physcally unnatural postons o lmbs when modellng human knematcs. Another speced model o a partcular area o nterest s ound n Reerence [7] where muscles are studed. A combned bomechancal model ncludng three rgd bones and actve muscles to smulate real human movements n a vertcal jump s studed n Reerence [3]. In order to very and perorm an nverse dynamcal problem, the moton o a lmb s captured usng cameras [8]. A number o artcles have been publshed on the analyss o the knematcs and moton o the entre human body, e.g. [9-11]. The natural coordnates approach has been appled n many cases when whole human body models are studed. Kraus et al. [1] have bult a human model ncludng over 1 degrees o reedom. In a model o ths sze, the number o parameters s hgh and they need to be systematcally determned. In reerence [12], the parameters area studed or the model appled n a vehcle crash smulaton. Crash test models have been under ntensve research and many studes are related to ths eld. In multbody applcaton topcs related to the computatonal technques are oten mportant and nterestng. In Reerence [13], the authors study the problem related to the

16 soluton o derental-algebrac equatons (DAE) n a multbody model denng an androd n a crash test. In all o the prevous studes, the bones were modelled as rgd bodes. Thereore, these models can not be used to calculate the bone strans. In ths study, the lower lmb bones are modelled as lexble bodes. Flexblty n multbody dynamc models can be taken nto account through several ways. In ths study, loatng rame o reerence [14-16] s used to account or the lexblty n the lower lmb model. 1.1 Scope o the Work Many parameters n the human body such as muscle orces and ther net moments orce about the anatomcal jonts are dcult to be measured through real experments. Thus, usng computer tools to model the human body becomes an mportant ssue. In ths work, a skeletal lower lmb s modelled usng the lexble multbody dynamc approach. The lexblty has been taken nto account n order to measure the strans n the bone, whch play a major role n strengthenng the bones. The strength o the bone s consdered an mportant resstant actor aganst the metabolc bone dseases. The man purpose o ths work s to show the capablty o the lexble multbody dynamc approach n modellng the human skeletal through measurng the strans n the bone. Not all the mportant ssues n human skeletal modellng are covered n ths work. Aspects, such as neural control system or muscle tendons modellng are not addressed here, as they have neglgble eects on the man purpose o ths work.

17 2. FLOATING FRAME OF REFERENCE FORMULATION In theory all the bodes are lexble and have nnte number o degree o reedom. However n practce there are number o ways to account or lexblty. In ths chapter, a method to take nto account the lexblty o the bodes n multbody dynamc system s explaned n detals. Ths method s called the loatng rame o reerence. In the ollowng sectons o ths chapter a detaled ormulaton o the equatons o moton based on the loatng rame o reerence or a planar lexble body s presented. For a better accuracy n modellng the elastc deormaton n the lexble body, the nte element method s used n the loatng rame o reerence ormulaton. Ths method that mples dscretzaton the lexble body nto elements and each element conssts o number o nodes s explaned n later sectons n ths chapter. Due to the hgh number o elastc coordnates resulted rom dscretzaton o the lexble body nto nte number o elements, modal reducton method s presented at the end o ths chapter to reduce the number o the elastc coordnates. 2.1 Descrpton o Knematcs The loatng rame o reerence ormulaton s based on the use o two coordnate systems; reerence and elastc coordnate systems. Fgure 2.1 shows the loatng rame o reerence coordnate systems, or a planar lexble body.

18 X 2 O X 2 u u o P u R r P X 1 Fgure 2.1. Coordnates or a planar lexble body n the loatng rame o reerence ormulaton. X 1 The generalzed coordnates system o the lexble body shown n the prevous gure can be expressed as ollows: [ q q ] q = (2.1) where r q r s the vector o reerence coordnates, whch descrbes the translaton and rotaton o the lexble body coordnate system 1 X 2 X wth respect to the global coordnate system X 1 X 2 and q s the vector o elastc coordnates, whch descrbes the elastc deormaton o the lexble body wth respect to the body coordnate system. The vectorq can be expressed as ollows: r [ R θ ] T r q = (2.2) where R s the vector that descrbes the translaton o the orgn O o the lexble body coordnate system wth respect to the global coordnate system andθ s the

19 orentaton angle o the lexble body coordnate system wth respect to the global coordnate system. The vector q n Eq 2.1 can be expressed as ollows: q [ q q L q ] T = (2.3) 1 2 n where n s the number o the elastc coordnates. The vector u shown n the prevous gure, descrbes the poston o any arbtrary pont P on the lexble body wth respect to the body coordnate system. The vector can be expressed as ollows: o u = u + u (2.4) o where u s the vector descrbes the undeormed poston o pont P wth respect to the body coordnate system and u s the vector descrbes the deormed poston o pont u P wth respect to the body coordnate system. The vector can be expressed by means o shape uncton matrx and elastc coordnates as ollows: u = S q (2.5) where S s a space dependent shape matrx, whch denty the shape o the deormaton or each pont n the lexble body. Substtutng Eq 2.5 nto Eq 2.4 yelds to the ollowng equaton: o u = u + S q (2.6) P The vector r shown n the prevous gure, descrbes the poston o pont P wth respect to the global coordnate system. The vector can be expressed n the ollowng equaton: P o r = R + A u = R + A ( u + S q ) (2.7) where A s the transormaton matrx, whch descrbes the rotaton o the lexble body coordnate system wth respect to the global coordnate system. It can be expressed as ollows: cosθ sn θ A = (2.8) sn θ cosθ

20 Descrpton o Velocty In order to get the velocty equatons, Eq 2.7 has to be derentated wth respect to the tme. Ths yelds to the velocty vector o any arbtrary pont on the lexble body wth respect to the global coordnate system, whch can be expressed as ollows: = & & A u& (2.9) r & P R + A u + where R & s the vector o translatonal veloctes o the orgn O o the lexble body wth respect to the global coordnate system and u& can be obtaned by derentatng Eq 2.6 wth respect to the tme. Ths yelds to the ollowng equaton: u & = S q& (2.1) whereq& s the velocty o the generalzed elastc coordnates. One may notce o here, that derentatng the vector u wth respect to the tme, yelds to zero as the vector s constant. The matrx θ A & n Eq 2.9 can be expressed as ollows: A & = A S θ& (2.11) whereθ & s the angular velocty o the lexble body coordnate system wth respect to the global coordnate system and A s the matrx results rom takng the dervatve o the transormaton matrx descrbed n Eq 2.8 wth respect to the rotaton angleθ. The matrx can be expressed as ollows: sn θ cosθ A θ = (2.12) cosθ sn θ Subsstng Eq 2.1 nto Eq 2.9 yelds to the ollowng equaton: = & & A S q& (2.13) r & P R + A u + The precedng equaton can be smpled or later dervaton purpose, by rewrtng the central term on the rght hand sde as ollows: & B θ & A u = (2.14) where B B can be expressed as ollows: θ = A u (2.15) Substtutng Eq 2.14 nto Eq 2.13 yelds to the ollowng equaton: θ

21 = & θ & A S & (2.16) r & P R + B + q The velocty vector dened n the precedng equaton can be descrbed n a parttoned orm as ollows: R& r& P = [ I B A S ] θ & (2.17) q& where I s the 2 x 2 dentty matrx. Equaton 2.17 can be also expressed as ollows: r & P = L q& (2.18) whereq& s the velocty o the generalzed coordnates system o the lexble body and L s a matrx whch can be expressed as ollows: [ I B A S ] L = (2.19) Descrpton o Acceleraton Recallng back Eq 2.18, and derentatng t wth respect to tme, the acceleraton o pont expressed as ollows: & r P L q& + P wth respect to the global coordnate system can be = & L q& (2.2) whereq& & s the acceleraton o the generalzed elastc coordnates and expressed as ollows: [ B& A& S ] L & can be & = (2.21) L where s a 2 x 2 zero matrx and B & can be obtaned by derentatng Eq 2.15 wth respect to the tme whch yelds to the ollowng equaton: B & = A u θ & + A S q& (2.22) θ 2.2 Knematc Constrants Multbody dynamc system conssts o several bodes connected to each other by means o jonts. These jonts restrct the system moblty because the moton o derent bodes s no longer ndependent. Consequently, the movement or the bodes n multbody dynamc system are related to each other by means o

22 constrants equatons. Constrants equatons can be taken nto account n the equatons o moton n multbody dynamc system usng two technques; Embeddng and Augmented technque. In ths study, the knematc constrant equatons are taken nto account usng augmented technque. Augmented equatons o moton ormulaton are based on the use o Lagrangan multplers. The knematc constrant equatons can be expressed n the general orm as ollows: C ( q, t) = (2.23) whereq s the vector o the total multbody system generalzed coordnates, t s the tme and C s the vector o lnearly ndependent constrant equatons o the multbody system. The number o degree o reedom (DOF) o a multbody system whch s equal to the ndependent generalzed coordnates can be expressed as ollows: n n c = DOF (2.24) wherens the number o generalzed coordnates o the multbody system and nc s the number o the constrant equatons o the system. Applyng a vrtual dsplacement to the knematc constrant equatons expressed n Eq 2.23 leads to the ollowng equaton: C q δ q = (2.25) wherec s the jacoban matrx. The jacoban matrx has ( x n) dmenson. q Consequently, the jacoban matrx has a ull row rank. Jacoban matrx can be obtaned by derentatng the algebrac constrant equatons wth respect to the generalzed coordnates o the multbody system. In general t can be expressed as ollows: n c C q 1 C1 / q 1 C2 / q = M 1 Cn / q c C C C 1 2 nc / q / q M 2 2 / q 2 L L M L C C C 1 2 nc / q / q M n n / q n (2.26)

23 2.3 Descrpton o Inerta The mass matrx o the lexble body shown n Fgure 2.1 can be dened usng derent components as the case n rgd body. The mass matrx can be expressed as ollows: M m RR = symmetrc m m Rθ θθ m m m R θ (2.27) where the term m RR represents the mass matrx assocated wth the translatonal coordnates o the lexble body coordnate system. It can be expressed as ollows: m m RR = ρ IdV = (2.28) m V where I s a 2 x 2 dentty matrx, ρ s the densty o the lexble body, V s the volume o the lexble body and m s the mass o the lexble body. The term m Rθ whch represents the couplng between the lexble body reerence translatonal and rotatonal coordnates can be expressed as ollows: [ I S q ] Rθ = Aθ 1 (2.29) m + where the vector I1denes the moment o the body mass about the axes o the body coordnate n the undeormed state. Consequently, the vector I1wll be equal to zero n case the orgn o the body coordnate s ntally attached to the center o the mass o the body. The vector I1can be expressed as ollows: I1 = ρ uodv (2.3) V In Eq 2.29 the vector S q represents the change n the moment o the mass due to the deormaton. In whch the matrx S can be wrtten as ollows: S = ρ u dv (2.31) V o The term m R whch represents the couplng between the translatonal reerence moton and the elastc deormaton, can be expressed as ollows:

24 R m = A S (2.32) The term m θ whch represents the couplng between the rotatonal reerence moton and the elastc deormaton, can be expressed as ollows: ~ T ~ mθ = ρ u o IS dv + q S (2.33) V where I ~ s a skew symmetrc matrx whch can be dented as ollows: ~ 1 I = A T θ A = (2.34) 1 And S ~ s a constant skew symmetrc matrx dented as ollows: ~ T ~ S = ρ S IS dv (2.35) V One may note that mθ conssts o two parts; the rst part s constant, whle the second one depends on the elastc coordnates o the lexble body. The term m whch s assocated wth the lexble body coordnates s a constant matrx. The matrx can be dened as ollows: m = ρ S S dv (2.36) V The term T m θθ conssts o summaton o three scalars dented as ollows: θθ = ( θθ ) rr + ( mθθ ) r + ( mθθ m m ) (2.37) The rst scalar s rr ( m θθ ) represents the mass moment o nerta o the lexble body, n the undeormed state about perpendcular axs to the plane. It can be dented as ollows: θθ ) rr ( m = ρ u u dv (2.38) V T o o The last two scalars ( m θθ ) r and m ) ( θθ represent the change n the mass moment o nerta o the lexble body due to deormaton. They can be dented respectvely as ollows: T T ( mθθ ) r = 2 ρ uo u dv = 2 ρ uo S dv q (2.39) V V T ( mθθ ) = q m q (2.4)

25 2.4 Generalzed Forces The generalzed orces n the loatng rame o reerence ormulaton are dvded nto two parts; the rst part s the generalzed external orces, whle the second part s the generalzed elastc orces. In the Fgure 2.2, a lexble body s subjected to an external orce F actng on an arbtrary pont P on the body. F X 2 O X 2 u u o P u X 1 R r P X 1 Fgure 2.2. Generalzed orces n the loatng rame o reerence ormulaton. Generalzed External Forces The vector o an external orce F actng on an arbtrary pont P on the lexble body shown n Fgure 2.2, can be constant, sprng, dampng or varable orce, or a combnaton o all them. The vrtual work o external orces can be expressed as ollows:

26 e e δ W = Q δq (2.41) where Qe s The vector o generalzed external orces assocated wth the generalzed coordnates o the lexble body. The vector can be expressed as ollows: Q e where T T T [ Q Q Q ] = (2.42) R θ QR s the vector o generalzed external orces assocated wth the translatonal reerence coordnates o the lexble body. The vector can be expressed as ollows: T R T Q = F (2.43) The mddle component at the mddle on the rght hand sde o Eq 2.42, s the vector o generalzed external orces assocated wth the rotatonal reerence coordnate o the lexble body. The vector can be expressed as ollows: T T Q θ = F B (2.44) The last component on the rght hand sde n Eq 2.42, s the vector o generalzed external orces assocated wth elastc coordnates o the lexble body. The vector can be expressed as ollows: T T Q = F A S (2.45) Generalzed Elastc Forces It s mportant to note the derence between the vrtual work done by the external orces and the elastc orces on the lexble body. It has been notced prevously rom Eq 2.4 that the vrtual work done by external orce actng on a lexble body s assocated wth the body generalzed coordnates. Consequently, the external orce appled to the lexble body, results wth a translatonal or rotatonal dsplacement, or both o them, whch are assocated wth reerence coordnates o the lexble body. Those types o dsplacement are called rgd body moton, as the lexblty o the body s not taken nto account. In addton the external orce appled to the lexble body results wth an elastc lnear deormaton assocated wth the generalzed elastc coordnates o the lexble body. Such deormaton aects the shape o the lexble body. It should be mentoned here that ths type o deormaton due to the body lexblty, may

27 result wth regardless to the exstence o external orce appled on the body. For example, the elastc deormaton o a ree rotatng beam, when the eect o gravty orces s neglected [17]. The vrtual work done by the elastc orces can be expressed as ollows: s = V T δ W δ dv (2.46) where ollows: s the stress vector and s the stran vector. The stran vector can be as = D u (2.47) where D s the stran-dsplacement matrx, whch s a matrx whose components are the dervatve o the shape unctons wth respect to the lexble body axes. Substtutng Eq 2.5 nto Eq 2.47, n order to descrbe the stran vector n terms o the generalzed elastc coordnates o the lexble body, yelds to the ollowng equaton: = D S q (2.48) Assumng that the materal o the lexble body s a lnear sotropc, Hooke s law whch relate the stress and stran by the ollowng lnear equaton can be appled as ollows: = E (2.49) wheree s the symmetrc matrx o the elastc coecents. Substtutng Eq 2.48 nto Eq 2.49 yelds to the ollowng equaton: = E D S q (2.5) Substtutng Eq 2.48 and Eq 2.5 nto Eq 2.46, yelds to the ollowng equaton: s = V T T δ W q ( D S ) E D S δq dv (2.51) Rearrangng the precedng equaton, usng the symmetrcal property o elastc coecents matrx and the act that the elastc coordnates o the lexble body depends only on tme, yelds to the ollowng equaton: T T δws = q ( D S ) E D S dv δq (2.52) V The precedng equaton can be wrtten as ollows:

28 s δw = q K δq (2.53) where K s the symmetrc postve dente stness matrx assocated wth the generalzed elastc coordnates o the lexble body. It can be dened rom Eq 2.52 as ollows: K = ( D S ) E D S dv (2.54) V T The vrtual work descrbed n Eq 2.53 can be wrtten n a parttoned orm as ollows: s δw = T T T [ R θ q ] K δr δθ δq (2.55) One may notce rom the prevous equaton, that the generalzed stness matrx assocated wth the generalzed coordnates o the lexble body can be dened as ollows: K = K (2.56) It s shown clearly rom the prevous equaton that the vrtual work o elastc orces due to the lexblty o the body, s assocated only wth the generalzed elastc coordnates o the lexble body. 2.5 Quadratc Velocty Vector The quadratc velocty vector o the lexble body has three components. The vector can be expressed as ollows: Q [( Q ) ( Q ) ( Q ] T v v R v θ v) = (2.57) The rst component s assocated wth the translatonal reerence coordnates o the lexble body. It denes the corols and centrugal orces assocated wth translatonal reerence coordnates. It can be expressed as ollows: θ ) 2θ& A S q& 2 ( Q v) R = ( ) A ( S q + I1 θ (2.58)

29 The second component s assocated wth the rotatonal reerence coordnate o the lexble body. It denes the corols orces assocated wth the rotatonal reerence coordnate. It can be dented as ollows: T ( Q v) = 2θ & θ q& ( m q + Io) (2.59) where o I o s dented as ollows: I = ρ S u dv (2.6) V T o The last component s assocated wth the elastc coordnates o the lexble body. It denes the centrugal and corols orces assocated wth the elastc degrees o reedom o the body. It can be expressed as ollows: Q & 2 m q I & ~ ( ) = ( θ ) ( + ) + 2θ S q& (2.61) v o 2.6 Reerence Condtons In order to dene a unque elastc dsplacement eld wth respect to the body coordnate system, the lexble body has to be modelled n such a way that the rgd body modes have be elmnated. To ths end a set o reerence condtons correspond to the selected model have to be appled on the elastc coordnates o the lexble body. The set o the reerence condtons has to be chosen careully as t denes the nature o the lexble body coordnate system [17, 18]. For example, the smply supported and ree-ree reerence condtons dene a loatng rame body coordnate system, the orgn o whch s not rgdly attached to a materal pont o the lexble body beam model. However, the two cantlever beams reerence condtons, dene a body xed coordnate system, the orgn o whch s rgdly attached to the geometrc center o the beam n the undeormed state. The reerence condtons can be appled by means o a constant matrx called Boolean reerence transormaton matrx, whose elements are ether zeros or ones. The uncton o ths matrx s to select the elastc coordnates, where the reerence condtons should be appled. One may notce that Eq 2.7 ncludes rgd body modes. Thus, to elmnate those modes, a set o reerence condtons have to be appled. In general, Eq 2.7 can be rewrtten ater applyng the reerence condtons as ollows: P o r r = R + A ( u + S B q ) (2.62)

30 where r B s the Boolean reerence transormaton matrx dened by Shabana n Reerence [14]. 2.7 Equatons o Moton Ater ormulatng the mass matrx, the jacoban matrx, the stness matrx, the vector o the external generalzed orces and the vector o the quadratc velocty, the equatons o moton can be ormulated usng the augmented technque based on usng Lagrange multples. For a planar lexble body n the multbody dynamc system, Lagrange equaton o moton can be expressed as ollows: T T T T T dt d Q C q q q = + & (2.63) where T s the knetc energy o the lexble body, s the vector o Lagrange multplers and Q s the vector o the generalzed elastc and external orces. The vector can be expressed as ollows: e Q K q Q + = (2.64) The rst two terms on the let hand sde o Eq 2.63 can be dened as ollows: v T T T T dt d Q M q q q = && & (2.65) Substtutng Eq 2.64 and Eq 2.65 nto Eq 2.63 yelds to the ollowng Lagrange equaton o moton: v e T Q Q C K q M q q + = + + & & b,...n 1,2 = (2.66) where b n s the total number o the bodes n the planar multbody dynamc system. The prevous equaton can be expressed n a parttoned orm as ollows: C C C q R K q R m m m m m m q R + + T T T R R I RR symmetrc θ θ θθ θ θ θ && && && (2.67) + = v v R v e e R e ) ( ) ( ) ( ) ( ) ( ) ( Q Q Q Q Q Q θ θ

31 Equaton 2.66 s a system o second order derental equatons that has ( n+ nc ) number o unknowns whle the number o equatons s n. Thereore, addtonal knematc constrant algebrac equatons descrbng the jonts between the bodes n the system are added. Those equatons have to be satsed at all tmes durng the moton. The addton o those constrant equatons leads to the ollowng set o DAE: & M q + K q C ( q, t) = + C T q = Q e + Q v = 1,2,...,n (2.68) b 2.8 Fnte Element Assemblng Procedure Fnte element procedure s used n the loatng rame o reerence ormulaton to descrbe the elastc deormaton o the lexble body wth ncreased accuracy. Moreover, exact modellng o rgd body moton assocated wth large translaton and rotaton can be obtaned, whle nntesmal rotatons are used as nodal coordnates. In the nte element loatng rame o reerence, the lexble body s dscretzed to a nte number o elements. Each element conssts o number o nodes results rom dscretzaton, whch denes the elastc deormaton o the element. The poston o each element s dened wth respect to the body coordnate system usng element coordnate system. The elastc deormaton o each element s dened usng nodal coordnates and space dependant element shape matrx. The nodal coordnates or each element are dented n terms o the total nodal coordnates o the lexble body usng Boolean transormaton matrx. Reerence condtons are appled at the boundary nodes o the lexble body n order to elmnate the rgd body modes. For more convenence, Fgure 2.3 shows the coordnate systems or a planar lexble body n nte element loatng rame o reerence.

32 j X 2 X 2 X 2 j O j X 1 X 1 O Fgure 2.3. Fnte element loatng rame o reerence coordnates. X 1 It can be notced rom Fgure 2.3 that the lexble body has been dscretzed to a nte number o elements. The orgn O o the element j coordnate system X j j 1 X 2 orgn j s rgdly attached to a pont on the element. Whle on the other hand, the O o the lexble body coordnate system X 1 X 2 s not necessarly has to be rgdly attached to a pont on the lexble body. All the postons o the elements have to be dened wth respect to the body coordnate system. Thereore, t serves as a reerence or all elements and expresses the connectvty between them. The orentaton o the lexble body coordnate system s dened wth respect to the global coordnate system usng the transormaton matrx dened n Eq 2.8. In the nte element loatng rame o reerence assemblng procedure two constant mappng matrces are needed to be dented. Those matrces do not change and are kept constant durng the smulaton. The rst matrx s a 2 x 2 transormaton matrx used to dene the orentaton o the element j coordnate system wth respect to the lexble body coordnate system. The matrx can be expressed as ollows:

33 j j j cosβ sn β C = j j (2.69) sn β cosβ where the subscrpt j reers to the element j n the lexble body and j β s the orentaton angle o the element j coordnate system wth respect to the lexble body coordnate system. The second matrx s called Boolean mappng transormaton matrx, whose elements are ether zeros or ones. Ths matrx s used to dene the vector o element j nodal coordnates n terms o the total vector o elastc nodal coordnates o the lexble body as ollows: j j T e = B e (2.7) where j e s the vector o the element j nodal coordnates, B s the Boolean mappng transormaton matrx whch s dened by Shabana n Reerence [14] and e s the vector o the total elastc nodal coordnates o the lexble body. Recallng Eq 2.7 the vector o nodal coordnates o the element j can be expressed as ollows: j j o j e = e + e (2.71) where and j eo denes the nodal coordnates o the element j n the undeormed state e j denes the elastc deormaton o the nodal coordnates o the element j. The assumed dsplacement eld o the nodal coordnates o the element j can be expressed n the element j coordnate system usng the shape uncton o the element j as ollows: j j j w = S e (2.72) where j S s the space dependant shape matrx o the element j. The assumed dsplacement eld o the nodal coordnates o the element j dened n the prevous equaton can be expressed wth respect to the lexble body coordnate system as ollows: j j j u = C w (2.73) where j u s the vector o the assumed dsplacement eld o the nodal coordnates o the element j dened wth respect to the lexble body coordnate system. Substtutng Eq 2.72 and Eq 2.7 nto Eq 2.73 yelds to the ollowng equaton: j j j j T u = C S B e (2.74) j

34 The precedng equaton can be wrtten n a more compact orm as ollows: j j T u = N e (2.75) where j j N can be descrbed rom Eq 2.74 as ollows: j j j N = C S B (2.76) The vector descrbes the poston o any arbtrary pont j P on the element j can be expressed wth respect to the global coordnate system usng the two constant mappng matrces as ollows: j P j r = R + A N e (2.77) where j rp s the vector descrbes the locaton or any arbtrary pont element j wth respect to the global coordnate system. j P on the 2.9 Modal Reducton Usng nte element method n the loatng rame o reerence ormulaton results wth a large number o generalzed elastc nodal coordnates o the lexble body. Ths s due to the lexble body dscretzaton nto nte number o elements whch leads to a large number o nodal degrees o reedom. Thus, descrbng the deormaton o the body, requres descrbng the deormaton or each node, whch leads to a long and expensve computaton. As a result, a modal reducton method can be adopted to reduce the generalzed elastc nodal coordnates. The reducton s acheved by elmnatng the hgh natural requency modes whch may carry lttle energy. Removng those types o vbraton modes assocated wth hgh natural requences prevent the body rom adoptng the partcular deormaton shape assocated wth ths mode. Vbraton modes wth hgh natural requences have deormaton shape, whch are not mportant and nterestng n practce, so elmnatng them wll not aect consderably the soluton. Modal reducton oers an ecent way to reduce the number o generalzed elastc nodal coordnates wth a mnmum eect n accuracy o the soluton. Modal reducton can be carred out by transerrng rom the physcal nodal coordnate system o the deormable body nto the modal elastc coordnates. Ths can be accomplshed by modal transormaton. Modal transormaton can be adopted usng the ollowng two steps. Frst step, s to solve the egenvalue equaton derved by assumng the

35 lexble body vbratng reely about a reerence conguraton. The equaton s dened as ollows: m & e + K e = (2.78) & where & e& s the acceleraton o the elastc nodal coordnates o the lexble body. A tral soluton or the precedng equaton whch has been suggested by (Clough and Penzen 1975; Shabana 1997) can be dened as ollows: j t a e ω e = (2.79) Substtutng the precedng equaton nto Eq 2.78 yelds to the ollowng equaton: 2 [ K ) m ] a = (ω k k k 1,2,...,nn = (2.8) The precedng equaton s called the standard egenvalue problem. Where nn s the number o the elastc nodal coordnates o the lexble body resulted rom dscretzaton, ωk s a set o egenvalues or natural requences assocated wth each nodal coordnate o the lexble body and ak are the correspondng egenvectors or the egenvalues. The egenvectors are sometmes called as the normal modes or the mode shapes. The second step s to elmnate the modes assocated wth hgh natural requences, n addton to the rgd boy modes. The modes that have zero natural requences and thus zero egenvalues are called rgd body modes. As a result o the second step, a reduced model wth m low requency mode shapes can be obtaned. It s mportant to note that the number o mode shapes results ater elmnaton m s much less than the number o the nodal coordnates n ). Ater choosng the mode shapes best descrbng the n ( m << nn deormaton o the body, a coordnate transormaton rom the physcal nodal coordnates to the modal elastc nodal coordnates can be accomplshed as ollows: e p (2.81) where and p s the vector o the modal elastc nodal coordnates o the lexble body s the modal transormaton matrx whose columns are the low requency m mode shapes. The matrx can be expressed as ollows: [ a a L a ] T = (2.82) 1 2 m

36 where am represents the mode shape correspond to the low natural requency ω m. The vector o the generalzed coordnates system o the lexble body can be expressed by means o the modal coordnate system as ollows: q p (2.83) where p s the vector o the modal coordnates o the lexble body and s the modal transormaton matrx or the total vector o the generalzed coordnates system o the lexble body. The precedng equaton can be rewrtten n a parttoned orm as ollows: q I r = e pr p (2.84) where p r s the vector o the modal reerence coordnates. It can be notced rom the prevous equaton that the reerence modal coordnates are equal to the physcal reerence coordnates. Orthogonalty o the Mode Shapes The orthogonalty o the mode shapes whch was proven by Shabana [15], s an mportant property. Ths property can be used to obtan a dagonal mass and stness matrces whch yeld to md number o uncoupled derental equatons. To ths end, Eq 2.68 can be rewrtten usng the vector o the modal coordnates o the lexble body as ollows: M p T & + K p + C = Qe + Q p v (2.85) Premultplyng the precedng equaton by T M p T yelds to the ollowng equaton: T T T & + K p + C = Qe + Q p v (2.86) where T M can be expressed as ollows: M p = T M a = T 1 T 1 M a M a T 2 T 2 M a M O a T m M M a T m (2.87)

37 = m m m m ˆ ˆ ˆ 2 1 M O M M where p M s the modal mass matrx and mˆ are the modal mass coecents. The matrx T K can be expressed as ollows: = = T m T m T T T T T p K a a K a a K a a K K M O M M (2.88) = m k k k 2 1 M O M where p K s called the modal stness matrx and k are the modal stness coecents. It can be notced rom Eq 2.87 and Eq 2.88 that the modal mass and stness matrces are dagonal as a result o the orthogonalty property o the mode shapes. The vectors on the rght hand sde o the Eq 2.86 can be expressed respectvely as ollows: e T e Q Q = (2.89) v T v Q Q = (2.9) It s mportant to note rom Eq 2.86 that the jacoban matrx s evaluated by derentatng the constrant equatons wth respect to the modal coordnates. However, t s more convenent to derentate the constrant equatons wth respect to the physcal coordnates. Ths can be accomplshed usng the ollowng relaton: C p q q C p C C p p = = = (2.91) Thereore the jacoban matrces assocated wth the modal reerence and elastc nodal coordnates can be expressed as ollows: p r q r C C = (2.92)

38 C p e = C (2.93) Solvng Eq 2.86 yelds to a number m o modal elastc nodal coordnates. Substtutng the soluton nto Eq 2.81 yelds to the physcal elastc nodal coordnates whch descrbe the deormaton o the body. Crag-Bampton Method The modal transormaton matrx expressed n Eq 2.82 contans the selected mode shapes that best descrbe the deormaton. However, to obtan acceptable dynamc accuracy, an excessve number o mode shapes may be stll requred. As a result, the modal reducton may lose ts mportance n decreasng the number o deormaton mode shapes whch are not nterestng. To overcome ths problem, one o the component mode synthess (CMS) methods, whch has been wdely used and s avalable n a number o commercal nte element codes such as ANSYS [19] can be appled. Ths method s called Crag-Bampton method [2]. Ths method can be appled by denng two sets o modes known as Crag- Bampton modes. The rst set s called constrant modes, whch can be obtaned by gvng each boundary DOF a unt dsplacement whle holdng all other boundary DOF xed. These modes span all possble motons o the boundary DOF. To llustrate the methodology o the constrant modes, Fgure 2.4 shows an example o two derent constrant modes or a beam that has attachment ponts at the two ends. Y Y X Fgure 2.4. Crag-Bampton constrant modes. X The prevous gure on the let, shows the constrant mode corresponds to a unt translaton, whle the gure on the rght, shows the constrant mode corresponds to a unt rotaton. The second set s called xed boundary normal modes, whch can be obtaned by xng the boundary DOF and computng an egensoluton. These modes span all possble motons o the nteror DOF. To llustrate the

39 methodology o the xed boundary normal modes, Fgure 2.5 shows two derent xed boundary normal modes or a beam that has attachment ponts at the two ends. Y Y X X Fgure 2.5. Crag-Bampton xed boundary normal modes. As a result o denng the prevous two sets o modes, a new set o coordnates, called modal coordnates o the Crag-Bampton modes, s ormed. Modal coordnates o the Crag-Bampton modes can be dened by means o the vector o generalzed coordnates system o the lexble body as ollows: q = pˆ (2.94) where s the Crag-Bampton transormaton matrx o the vector o generalzed coordnates system o the lexble body and pˆs the vector o modal coordnates o the Crag-Bampton modes. The precedng equaton can be rewrtten n a more explct orm as ollows: q I r = e IC IN pˆ C pˆ N (2.95) where IC s the Crag-Bampton transormaton matrx o the vector o elastc nodal coordnates o the lexble body or the neteror DOF n the constrant modes, IN s the Crag-Bampton transormaton matrx o the vector o elastc nodal coordnates o the lexble body or the neteror DOF n the normal modes, pˆ C s the vector o modal coordnates o the Crag-Bampton constrant modes o the lexble body and pˆ N s the vector o modal coordnates o the Crag- Bampton xed boundary normal modes o the lexble body. The Crag- Bampton modes are not orthogonal. Thereore, the Crag-Bampton transormaton matrx s unsutable or drect use n dynamc system smulaton, as t can not be

40 utlzed to obtan dagonal mass and stness matrces. To overcome ths problem, a new transormaton matrx Nˆ whch transorms the Crag-Bampton modes to equvalent orthogonal modes expressed by means o modal coordnates s dened. The new transormaton matrx can be obtaned by solvng the ollowng egenvalue problem: [ K ˆ ( ) Mˆ ] 2 = where λ (2.96) Kˆ s the Crag-Bampton stness transormaton matrx, whch can be expressed by means o the generalzed stness matrx o the lexble body as ollows: ˆ T (2.97) K = K λ n Eq 2.96 s a set o egenvalues or natural requences assocated wth each modal coordnate o the Crag-Bampton modes o the lexble body, correspondng orthogonal egenvectors or the egenvalues and are the Mˆ s the Crag- Bampton mass transormaton matrx, whch can be expressed by means o the generalzed mass matrx o the lexble body as ollows: ˆ T (2.98) M = M The orthogonal egenvectors result rom solvng the egenvalue problem expressed n Eq 2.96 are arranged n the transormaton matrx Nˆ, whch transorms the modal coordnates o the Crag-Bampton modes to an equvalent orthogonal modal coordnates. Consequently, the modal coordnates can be expressed by means o modal coordnates o the Crag-Bampton modes as ollows: p = Nˆ pˆ (2.99) Usng Eq 2.83 and Eq 2.99 the eect o the superposton can be expressed as ollows: q = p = Nˆ pˆ (2.1) The Crag-Bampton transormaton matrx o the vector o generalzed coordnates system o the lexble body can be expressed usng the prevous equaton as ollows: = Nˆ (2.11)